Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#2249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8)(9,17)(10,22)(11,14)(12,25)(13,19)(15,18)(16,23)(20,29)(21,36)(24,37)(26,35)(27,38)(28,33)(30,32)(31,34);(7,9,18,32,12,21,24,35,11,20,33,38,23,34,10,19)(8,13,22,31,16,27,28,29,14,26,37,36,25,30,15,17);(3,4,5,6)(8,14)(13,26)(15,28)(16,25)(17,29)(22,37)(27,30)(31,36)76 conjugacy classes — view the full character table.
| maximal subgroup of | #141547, #141548, C2×#7446, … |
| maximal normal subgroup of | #141547, #141548, C2×#7446, … |
| maximal quotient of | C2×#12503, C2×#12504, C2×#12505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26700);